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Easier algebra with exponents
CLASS WORK
1 DEFINITION
2 ^{3} = 2 × 2 × 2 and a ^{4} = a × a × a × a and b × b × b = b ^{3}
also
(a+b) ^{3} = (a+b) × (a+b) × (a+b) and ${\left(\frac{2}{3}\right)}^{4}=\left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)$
1.1 Write the following expressions in expanded form:
4 ^{3} ; (p+2) ^{5} ; a ^{1} ; (0,5) ^{7} ; b ^{2} × b ^{3} ;
1.2 Write these expressions as powers:
7 × 7 × 7 × 7
y × y × y × y × y
–2 × –2 × –2
(x+y) × (x+y) × (x+y) × (x+y)
1.3 Answer without calculating: Is (–7) ^{6} the same as –7 ^{6} ?
–5 ^{2} and (–5) ^{2} –12 ^{5} and (–12) ^{5} –1 ^{3} and (–1) ^{3}
a ^{r} = a × a × a × a × . . . (There must be r a’s, and r must be a natural number)
2 ^{2} = 4; 2 ^{3} = 8; 2 ^{4} = 16; etc. 3 ^{2} = 9; 3 ^{3} = 27; 3 ^{4} = 81; etc. 4 ^{2} = 16; 4 ^{3} = 64; etc.
Most problems with exponents have to be done without a calculator!
2 MULTIPLICATION
2.1 Simplify: (don’t use expanded form)
7 ^{7} × 7 ^{7}
(–2) ^{4} × (–2) ^{13}
( ½ ) ^{1} × ( ½ ) ^{2} × ( ½ ) ^{3}
(a+b) ^{a} × (a+b) ^{b}
a ^{x} × a ^{y} = a ^{x+y} also ${a}^{x+y}={a}^{x}\times {a}^{y}={a}^{y}\times {a}^{x}$ , e.g. ${8}^{\text{14}}={8}^{4}\times {8}^{\text{10}}$
3 DIVISION
3.1 Try these: $\frac{{a}^{6}}{{a}^{y}}$ ; $\frac{{3}^{\text{23}}}{{3}^{\text{21}}}$ ; $\frac{{\left(a+b\right)}^{p}}{{\left(a+b\right)}^{\text{12}}}$ ; $\frac{{a}^{7}}{{a}^{7}}$
Also ${a}^{x-y}=\frac{{a}^{x}}{{a}^{y}}$ , e.g. ${a}^{7}=\frac{{a}^{\text{20}}}{{a}^{\text{13}}}$
4 RAISING A POWER TO A POWER
4.1 Do the following:
5 THE POWER OF A PRODUCT
(2a) ^{3} = (2a) × (2a) × (2a) = 2 × a × 2 × a × 2 × a = 2 × 2 × 2 × a × a × a = 8a ^{3}
5.1 Do these yourself: (4x) ^{2} ; (ab) ^{6} ; (3 × 2) ^{4} ; ( ½ x) ^{2} ; (a ^{2} b ^{3} ) ^{2}
6 A POWER OF A FRACTION
6.1 Do these, but be careful: ${\left(\frac{2}{3}\right)}^{p}$ ${\left(\frac{\left(-2\right)}{2}\right)}^{3}$ ${\left(\frac{{x}^{2}}{{y}^{3}}\right)}^{2}$ ${\left(\frac{{a}^{-x}}{{b}^{-y}}\right)}^{-2}$
end of CLASS WORK
TUTORIAL
1. $\frac{{a}^{5}\times {a}^{7}}{a\times {a}^{8}}$ 2. $\frac{{x}^{3}\times {y}^{4}\times {x}^{2}{y}^{5}}{{x}^{4}{y}^{8}}$
3. ${\left({a}^{2}{b}^{3}c\right)}^{2}\times {\left({\text{ac}}^{2}\right)}^{2}\times {\left(\text{bc}\right)}^{2}$ 4. ${a}^{3}\times {b}^{2}\times \frac{{a}^{3}}{a}\times \frac{{b}^{5}}{{b}^{4}}\times {\left(\text{ab}\right)}^{3}$
5. $\left(2\text{xy}\right)\times {\left({\mathrm{2x}}^{2}{y}^{4}\right)}^{2}\times \left(\frac{{\left({x}^{2}y\right)}^{3}}{{\left(2\text{xy}\right)}^{3}}\right)$ 6. $\frac{{2}^{3}\times {2}^{2}\times {2}^{7}}{8\times 4\times 8\times 2\times 8}$
end of TUTORIAL
Some more rules
CLASS WORK
1 Consider this case: $\frac{{a}^{5}}{{a}^{3}}={a}^{5-3}={a}^{2}$
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